where Anl are the normalization constants, Rnl(r) are the radial energy eigenfunctions, and Ylm(θ,φ) are the spherical harmonics.

In this Exploration, Animation 1 depicts ψnlm in the zx plane only. In Animation 2, Φm(φ), Plm (θ), R(r), and ψnlm are visualized in one of four panels. The entire solution, ψnlm, is visualized in the zx plane only. To generate the spherical energy eigenfunction, first imagine the rotation of ψnlm about the z axis; this gives you the shape of the energy eigenfunction. Then, to get the phase of the energy eigenfunction, project the phase (color) from Φm(φ).

For a given value of n and l, how does the number of angular lobes in Plm (θ) change with m?

For a given value of n and l, how does the number of wavelengths (from blue to blue is one wavelength) in Φm(φ) change with m?

How do the non-zero l values affect the radial energy eigenfunction?

How do the non-zero m values affect the radial energy eigenfunction?

For n = 3, how many times does the radial energy eigenfunction cross zero (change signs) for each possible value of l? Try this for a few other values for the principal quantum number, n, and see if your conclusion holds.

Note: Right click on any applet to make a copy of the image. The mouse coordinates may be observed by left-clicking within the graph.